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Structures of left n-invertible operators and their applications

Caixing Gu (2015)

Studia Mathematica

We study left n-invertible operators introduced in two recent papers. We show how to construct a left n-inverse as a sum of a left inverse and a nilpotent operator. We provide refinements for results on products and tensor products of left n-invertible operators by Duggal and Müller (2013). Our study leads to improvements and different and often more direct proofs of results of Duggal and Müller (2013) and Sid Ahmed (2012). We make a conjecture about tensor products of left n-invertible operators...

Su un ampliamento della teorìa degli operatori lineari invarianti rispetto ad un gruppo di congruenze

Lucilla Bassotti Rizza (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let A be an open subset of n , W m ( A ) the linear space of m -vector valued functions defined on A , G { γ } a group of orthogonal matrices mapping A onto itself and T { T γ } a linear representation of order m of G . A suitable group 𝒯 ( G , T ) of linear operators of W m ( A ) is introduced which leads to a general definition of T -invariant linear operator with respect to G . When G is a finite group, projection operators are explicitly obtained which define a "maximal" decomposition of the function space into a direct sum of subspaces...

Submultiplicative functions and operator inequalities

Hermann König, Vitali Milman (2014)

Studia Mathematica

Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality” T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ). Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form Tf = ⎧ ( H f / H ) f ' p , f’ ≥ 0, ⎨ ⎩ - A ( H f / H ) | f ' | p , f’ < 0, with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions...

Subnormal operators, cyclic vectors and reductivity

Béla Nagy (2013)

Studia Mathematica

Two characterizations of the reductivity of a cyclic normal operator in Hilbert space are proved: the equality of the sets of cyclic and *-cyclic vectors, and the equality L²(μ) = P²(μ) for every measure μ equivalent to the scalar-valued spectral measure of the operator. A cyclic subnormal operator is reductive if and only if the first condition is satisfied. Several consequences are also presented.

Subnormality and cyclicity

Franciszek Hugon Szafraniec (2005)

Banach Center Publications

For an unbounded operator S the question whether its subnormality can be built up from that of every S f , the restriction of S to a cyclic space generated by f in the domain of S, is analyzed. Though the question at large has been left open some partial results are presented and a possible way to prove it is suggested as well.

Subsets of nonempty joint spectrum in topological algebras

Antoni Wawrzyńczyk (2018)

Mathematica Bohemica

We give a necessary and a sufficient condition for a subset S of a locally convex Waelbroeck algebra 𝒜 to have a non-void left joint spectrum σ l ( S ) . In particular, for a Lie subalgebra L 𝒜 we have σ l ( L ) if and only if [ L , L ] generates in 𝒜 a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.

Subspaces with a common complement in a Banach space

Dimosthenis Drivaliaris, Nikos Yannakakis (2007)

Studia Mathematica

We study the problem of the existence of a common algebraic complement for a pair of closed subspaces of a Banach space. We prove the following two characterizations: (1) The pairs of subspaces of a Banach space with a common complement coincide with those pairs which are isomorphic to a pair of graphs of bounded linear operators between two other Banach spaces. (2) The pairs of subspaces of a Banach space X with a common complement coincide with those pairs for which there exists an involution...

Suitable domains to define fractional integrals of Weyl via fractional powers of operators

Celso Martínez, Antonia Redondo, Miguel Sanz (2011)

Studia Mathematica

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the L p ( ) -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative...

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

Let Y be a Banach space and let S L p be a subspace of an L p space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to S ( Y ) L p ( Y ) . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and e - t B is a positive contraction on L p for any...

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